#27296143

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅₃×#1595 ⋊ C₂
Sylow	C₅₃ ⋊ #12094

Subgroups

	subgroup	G/H
center	C₂²	#16810809
commutator, cosocle	C₂×C₂²⋊C₄×C₅₃	C₂³
Frattini	C₂×C₂²⋊C₄	C₂²×D₅₃
Fitting	C₅₃×#1595	C₂
socle	C₂²×C₅₃	B(4,𝔽₂)
hypercenter	#1595	D₅₃
hypocenter	C₅₃	#12094
LCS	G ▷ C₂×C₂²⋊C₄×C₅₃ ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₂²⋊C₄ ◁ #1595

570 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 77 points:

(9,11,13,17)(10,15,16,24)(12,19,20,18)(14,21,22,23)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77);(2,5)(3,8)(6,7)(10,14)(11,17)(15,23)(16,22)(18,19)(21,24)(25,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27);(1,2)(3,5)(4,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,24)(18,21)(19,23)(20,22)

Polycyclic presentation:

2^8,53|c^2=h,d^2=f,b^a=bd,c^a=ce,d^a=df,g^a=gh,c^b=ch,d^b=df,e^b=egh,d^c=dg,i^c=i^52,e^d=eh

Conjugacy classes & character table

623 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 81619);